Edited by author 16.04.2018 19:00

Optimal point is O.
1) O=A or O=B or O=C
2) We have triangle ABC with 0<A,B,C<pi;
   We know cos(AOB), cos(AOC), cos(BOC).
   Let z1=(c3*c3-c1*c1-c2*c2)/(2*c1*c2), z2=(c2*c2-c1*c1-c3*c3)/(2*c1*c3),
       z3=(c3*c3-c1*c1-c2*c2)/(2*c1*c2).

   If |z1|>1 or |z2| >1 or |z3|>1 => optimal point is A or B or C
   Let |zi|<=1, i=1..3
   teta=AOB.
   Theorem sin: BO/sin(teta) = AB/sin(AOB) = AO/sin(teta+AOB) = k1 and
                   CO/sin(A-teta) = AC/sin(AOC) = k2.
   =>AO=k1*sin(teta+AOB), BO=k1*sin(teta), CO=k2*sin(A-teta).
    (*) We want find min{f(teta)=c1*AO+c2*BO+c3*CO} on 0<teta<A.
    Derivative f: cos(teta)*[c1*k1*cos(AOB)+c2*k1-c3*k2*cos(A)]+sin(teta)*[-c1*k1*sin(AOB)-c3*k2*sin(A)]=0 or cos(teta)*a+sin(teta)*b=0, cos(teta)=+=sqrt(sqr(b)/(sqr(a)+sqr(b)).
    Then we check 0<teta<A and update ans.


Where is mistake?
I think mistake is (*) but I can't understand why.


P.S. We can use Theorem cos with BOC and get equation with teta, but this equation is big and not obviously that we find solution.

P.S. Your code not help me.

Edited by author 16.04.2018 19:08